Magnetized Four-Dimensional Z2 × Z2 Orientifolds

UNIVERSITA` DEGLI STUDI DI ROMA \TOR VERGATA" FACOLTA` DI SCIENZE MATEMATICHE, FISICHE E NATURALI Dipartimento di Fisica Magnetized four-dimensional Z Z Orientifolds 2 × 2 Tesi di dottorato di ricerca in Fisica Marianna Larosa Relatore Dott. Gianfranco Pradisi Coordinatore del dottorato Prof. Piergiorgio Picozza Ciclo XVII Anno Accademico 2004-2005 ii Contents 1 Introduction 3 1.1 The Bosonic String . 3 1.2 The Superstring . 5 1.3 Compactifications and Dualities . 9 1.4 Outline . 12 2 String Perturbation Theory 13 2.1 One Loop Partition Functions . 18 2.2 Toroidal Compactifications . 23 2.3 Orbifold Compactifications . 25 3 Type I Superstrings 29 3.1 Open Descendants or Orientifolds . 29 3.2 D-Branes . 34 3.3 Shift-Orbifolds . 38 3.4 Shift-Orientifolds . 39 3.4.1 NS-NS Bab and Shifts . 40 3.5 Z Z Orientifolds . 42 2 × 2 3.5.1 Z Z Models and Discrete Torsion . 42 2 × 2 3.5.2 Z Z Shift-orientifolds and Brane Supersymmetry . 46 2 × 2 4 Magnetic Deformations 51 4.1 The bosonic string in a uniform magnetic field . 52 4.2 Open Strings on a Magnetized Torus . 54 4.3 Open Strings on Magnetized Orbifolds . 60 4.4 Spectra of the [T 2(H ) T 2(H )]=Z orientifolds . 62 2 × 3 2 5 Magnetized D=4 Models 65 5.1 Magnetized Z Z Orientifolds . 66 2 × 2 iii iv CONTENTS 5.2 Magnetized Z Z Shift-orientifolds . 72 2 × 2 5.2.1 w2p3 Models . 72 5.2.2 w1w2p3 Models . 79 5.3 Brane Supersymmetry Breaking . 81 5.4 Conclusions . 83 A Lattice Sums in the Presence of a Quantized Bab 85 B Characters for the T 6=Z Z Orbifolds 87 2 × 2 C Massless Spectra 89 C.1 Closed Spectra of the Z Z orbifolds . 90 2 × 2 C.2 Open spectra of the Z Z orientifolds with ! = +1 . 92 2 × 2 i C.3 Open Spectra of the Magnetized Z Z Orientifolds with ! = +1 . 94 2 × 2 i C.4 Oriented Closed Spectra of the Z Z Shift-orientifolds . 96 2 × 2 C.5 Unoriented Closed Spectra of the p3 Models . 97 C.6 Orientifolds of the w2p3 Models . 98 C.7 Orientifolds of the w1w2p3 Models . 101 C.8 Non-chiral Orientifolds . 103 C.9 w2p3 Models with Brane Supersymmetry Breaking . 107 C.10 Open Spectra of the Undeformed Z Z Shift-orientifolds . 109 2 × 2 Acknowledgments This thesis is based on the work developed at the Physics Department of the Università di Roma \Tor Vergata" in collaboration with Dr. Gianfranco Pradisi, that I would like to thank for the support, the availability and friendness he gave to me. I am also very grateful to all the members of the Theoretical Group of the Physics Department, and in particular to Prof. Augusto Sagnotti, who introduced me to the study of String Theory, encouraging me with precious suggestions and stimulating discussions. Finally, I would like to thank all the staff of the Physics Department. 1 to Salvo 2 Chapter 1 Introduction It has been about fourty years since the era of String Theory [1] began with the seminal work of Veneziano [2] on the four hadron scattering amplitudes. The non-renormalizability of the Albert Einstein General Relativity [3, 4, 5, 6, 7, 8, 9, 10] has actually provided the main hint to re- consider String Theory as a possible candidate embracing the gravitational interaction into the frame of a unified quantum theory containing all known particles and fundamental interactions [11, 12]. In particular, the solution to the problem of the short-distance divergences of the quantum gravity and the unavoidable presence of spin two massless particles in the spectrum, have led to think of String Theory as a promising and finite quantum theory of gravity. Conceiving a theory of strings as a model for the description of nature, had the immediate consequence of giving up the fundamental ingredient of the quantum field theory, the point particle interpretation. 1.1 The Bosonic String The underlying idea on which String Theory rests is essentially an extension of the basic principle of Quantum Field Theory, to the case in which the fundamental entities are one-dimensional objects with characteristic length ls. In this context particles are waves propagating along the string, and the point-like excitations of field theory are replaced by one-dimensional excitations interacting in a geometrical way. Thus, in a D-dimensional space-time, a string sweeps out two- dimensional surfaces, or world-sheets Σ, parametrized by the two conventional variables τ and σ (τ ( ; + ); σ [0; π]) and described by the position of the string X µ(τ; σ) (µ = 0; ::; D 1). 2 −∞ 1 2 − Therefore, we are naturally led to study two-dimensional field theories. The simplest model of relativistic string is the bosonic string. Its motion in a flat D- dimensional space-time is defined by the following action [13, 14] 1 S = T d2ξ( p ggαβ(ξ)η @ Xµ@ Xν ); (1.1) − 2 − µν α β Z 3 1 usually known as Polyakov action, where T = 2πα0 is the string tension, α0 is the Regge slope 2 2 2 ([T ] = L− = M ), d ξ = dτdσ, and gαβ(ξ) (α, β = 0; 1) is the metric on the world-sheet. This action, describing D massless scalar fields coupled to gravity in two-dimensions, is the most convenient action for the quantization procedure, since it is invariant under: 1. global D-dimensional Poincarè transformations µ µν µ δX = Ω Xν + a ; (1.2) δgαβ = 0 ; (1.3) where Ωµν is an antisymmetric matrix of the D-dimensional representation of the SO(1; D − 1) group and aµ is a constant vector 2. local world-sheet diffeomorphisms (diff): • µ α µ δX = ζ @αX ; (1.4) δg = ζ + ζ ; (1.5) αβ 5α β 5β α i.e. it is independent from the choice of the coordinates ξα . Weyl rescalings of the metric on the world-sheets • δXµ = 0 ; (1.6) δgαβ = 2Λ(ξ)gαβ : (1.7) Actually there is a residual symmetry of the world-sheet theory, left over by the gauge-fixing of (diff Weyl)-invariance and that is fundamental for the construction of a consistent quantum × theory. It is conformal invariance, a key property of the simple action of eq. (1.1), that allows to get rid of negative norm states of the Hilbert space and thus must be preserved by quantum corrections. Conformal anomaly cancellation is thus a consistency condition of string theory. It reflects itself in the exact vanishing of the central charge imposing stringent restrictions on the dimension of the space-time, fixed to D = 26 for the bosonic string case. According to the possible choices of the boundary conditions, strings can come in two very distinctive varieties: closed or open. In particular, if the coordinate X µ satisfy periodic boundary conditions, Xµ(τ; σ) = Xµ(τ; σ + π), the strings are topologically equivalent to circles. By contrast, for open strings the endpoints are no more coincident, but can be free to move, and 0 0 thus subjected to Neumann boundary conditions X µ(τ; 0) = X µ(τ; π) , or fixed in space-time, i.e. such to satisfy Dirichlet boundary conditions X µ(τ; 0) = aµ ; Xµ(τ; π) = bµ ; with aµ and bµ constant vectors. 4 The quantization of the theory can follow several different methods, the simplest of which is the so called light-cone gauge quantization [15]. It consists in eliminating two degrees of freedom of the gauge-fixed theory, with the end result that only the transverse space-time directions are allowed to oscillate. As a consequence, are just the corresponding transverse vibrating modes that give rise to the complete spectrum of the theory. Both closed and open string theories present a tachyon at the lowest mass level, a particle with negative mass-squared. As it happens in field theory, this signals a wrong identification of the vacuum. The next mass level consists of massless bosonic space-time fields, that for the closed string are the transverse modes of a two tensor, for a total of (D 2)2 states, while for the open string they are the transverse modes of a − vector, i.e. (D 2) states. The massless closed spectrum thus describes: a traceless symmetric − tensor that can be identified with the graviton gij, a spin two massless particle whose presence made string theory the first candidate able to incorporate gravity in a unifying quantum field theory of all interactions; an antisymmetric tensor Bij and a scalar field, usually called the dilaton φ and whose vacuum expectation value is the string coupling constant. Beyond this level, one can find a tower of states with higher and higher masses and spins [2, 16, 17, 18]. Their typical mass is conventionally of the order of the Planck scale. This clearly reveals that, even if they are essential for the soft behavior of the string amplitudes at high-energy [19], they are marginal with respect to the low-energy physics description, which is instead dominated by the massless vibrational modes. It is worth to stress that, in contrast to closed strings, the presence of the two ends allows open strings to carry gauge degrees of freedom, known as Chan-Paton factors [20]. They are the charges of an internal symmetry, giving rise to non-abelian groups of the type U(n) in the oriented case and SO(n) or USp(n) in the unoriented case, but not of the Exceptional type of the classical Cartan classification [22, 23].

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